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Type IIA D-Branes, K-Theory and Matrix Theory c 1998 International Press Adv. Theor. Math. Phys. 2 (1998) 1373–1404 Type IIA D-Branes, K-Theory and Matrix Theory Petr Hoˇrava a a California Institute of Technology, Pasadena, CA 91125, USA [email protected] Abstract We show that all supersymmetric Type IIA D-branes can be con- structed as bound states of a certain number of unstable non-supersym- metric Type IIA D9-branes. This string-theoretical construction demon- strates that D-brane charges in Type IIA theory on spacetime manifold X are classified by the higher K-theory group K−1(X), as suggested recently by Witten. In particular, the system of N D0-branes can be obtained, for any N, in terms of sixteen Type IIA D9-branes. This sug- gests that the dynamics of Matrix theory is contained in the physics of magnetic vortices on the worldvolume of sixteen unstable D9-branes, arXiv:hep-th/9812135 v4 14 May 1999 described at low energies by a U(16) gauge theory. e-print archive: http://xxx.lanl.gov/abs/hep-th/9812135 1374 TYPE IIA D-BRANES, K-THEORY, AND MATRIX THEORY 1 Introduction When we consider individual D-branes in Type IIA or Type IIB string theory on R10, we usually require that the branes preserve half of the original su- persymmetry, and that they carry one unit of the corresponding RR charge. These requirements limit the D-brane spectrum to p-branes with all even values of p in Type IIA theory, and odd values of p in Type IIB theory. Once we relax these requirements, however, we can consider Dp-branes with all values of p. In Type IIA theory, we can consider p-branes with p odd, and in particular, a spacetime-filling 9-brane. All these states are non- supersymmetric unstable excitations in the corresponding supersymmetric string theory. Indeed, there is always a tachyon in the spectrum of the open string connecting one such Type IIA (2p 1)-brane to itself. Thus, such − D-brane configurations (and their counterparts on spacetimes of non-trivial topology) are highly unstable, and one expects that they should rapidly decay to the supersymmetric vacuum, by a process that involves tachyon condensation on the worldvolume. This is of course in perfect agreement with the field content of the corresponding low-energy supergravity in spacetime – there are no RR fields that could couple to any conserved charges carried by such non-supersymmetric branes. This does not seem to leave much room for surprises, but in fact, the full story is much more interesting. Configurations of unstable D-branes can sometimes carry lower-dimensional D-brane charges, and therefore, when the tachyon rolls down to the minimum of its potential and the state decays, it can leave behind a supersymmetric state that differs from the vacuum by a lower-dimensional D-brane charge – in other words, the state decays into a supersymmetric D-brane configuration. Typically, one can then represent the supersymmetric D-brane state as a bound state of the original system of unstable D-branes. This setup generalizes a special case studied in [1, 2], where one starts with an unstable configuration of an equal number of stable p-branes and stable anti-p-branes (or p-branes for short), and finds lower-dimensional sta- ble D-brane states as bound states in this system. The p-brane p-brane system is unstable, and the instability manifests itself by the presence of a tachyon in the spectrum of the p-p open strings at brane separations shorter than the string scale [3]-[7]. In Type IIB or Type I string theory, one can use this construction to represent any stable D-brane state as a bound state of a certain number of 9-brane 9-brane pairs [1, 2]. In some cases, the unstable non-supersymmetric state decays into a stable PETR HORAVAˇ 1375 state that is not supersymmetric, but is protected from further decay by charge conservation. One typical example of such states is the SO(32) spinor of Type I theory [1], which is non-supersymmetric but stable, since it is the lowest spinor state in the theory. Such non-supersymmetric D-branes can be found using a direct boundary state construction [8] , or alternatively as bound states of p-brane p-brane pairs [1] . This remarkable bound-state construction was discovered by Sen [1] (following some earlier work in [9] ), and was further systematized by Witten [2] . It turns out that the proper setting for all conserved D-brane charges in general compactifications is, in fact, K-theory [10, 2]. (Some early indications of possible connections between D-branes and K-theory can be found in [11]-[18].) In this paper, we generalize the construction that uses unstable configura- tions of pairs of stable branes, and consider bound states in general unstable non-supersymmetric configurations of D-branes such as the unstable 9-brane discussed above, where the individual D-branes are no longer required to be stable. Our motivation for this generalization will be clear from the following example. We will be interested in stable D-branes in Type IIA string theory, for simplicity in R10. In Type IIB theory, we can in principle construct any stable D-brane state as a bound state of a certain number of 9-brane 9-brane pairs wrapping the whole spacetime. It is certainly desirable to have, on the Type IIA side, a similar construction that would enable us to study stable D-branes as bound states in unstable configurations of branes of maximal di- mension. However, there are no stable D9-branes in Type IIA string theory! While we can indeed represent any stable D-brane of Type IIA as a bound state of an 8-brane 8-brane system, this construction requires a preferred choice of a submanifold of codimension one in space-time, representing the worldvolume of the 8-brane 8-brane system. Therefore, it breaks some of the spacetime symmetries that we want to keep manifest in the theory, and limits the kinematics of branes that can be studied this way. One of the main points of this paper is to present a string-theoretical construction that keeps all spacetime symmetries manifest. This construction enables us to consider any stable D-brane of Type IIA theory as a bound state of a system of the unstable 9-branes discussed above. In fact, this construction turns out to be intimately related to the statement that D-brane charges in Type IIA string theory are classified by the higher K-theory group K−1(X) of the spacetime manifold X, suggested recently by Witten in [2] . This paper is organized as follows. In section 2 we briefly review the relation of Type IIB D-brane charges 1376 TYPE IIA D-BRANES, K-THEORY, AND MATRIX THEORY and bound-state constructions to K-theory, and preview the Type IIA case. In section 3.1 we introduce the unstable 9-brane of Type IIA string the- ory. General 9-brane configurations wrapping spacetime manifold X are studied in section 3.2. We argue that inequivalent configurations of 9-branes – modulo 9-branes that can be created from or annihilated to the vacuum – are classified by the higher K-theory group K−1(X). In section 3.3 we show that any given stable D-brane configuration of Type IIA string the- ory can be represented as a bound state of a certain number of unstable Type IIA 9-branes. In the worldvolume of the 9-branes, the bound state appears as a stable vortex in the tachyon field, accompanied by a non-trivial gauge field carrying a generalized magnetic charge. In the particular case of bound states in codimension three, this precisely corresponds to the ’t Hooft- Polyakov magnetic monopole. We generalize our discussion to the case of Type I′ theory in section 3.4, and argue that Type I′ D-brane charges are similarly classified by the Real K-theory group KR−1(X). In section 4 we focus on possible implications of our construction to Ma- trix theory [19] . We use our 9-brane bound-state construction to study a general system of N D0-branes in Type IIA theory. First we show that a D0-brane can be constructed as a bound state of sixteen unstable D9- branes. The low-energy worldvolume theory on the 9-branes is a certain non-supersymmetric U(16) gauge theory, with a tachyon in the adjoint repre- sentation of U(16). In this worldvolume theory, the D0-brane is represented as a topologically stable vortex-monopole configuration of the tachyon and the gauge field. Multiple D0-brane configurations are in general also de- scribed by sixteen 9-branes, and appear as multi-vortex configurations in the U(16) gauge theory on the spacetime-filling worldvolume. This construction thus leads to the intriguing possibility that the dynamics of Matrix theory – as described by a particular limit of the system of N D0-branes of Type IIA string theory – can be contained in the dynamics of vortices on the worldvol- ume of a fixed system of sixteen D9-branes, with the individual D0-branes represented by vortices in the worldvolume field theory. This construction exhibits some striking similarities with the holographic field theory of [20] . While this paper was being finished, another paper appeared [21] whose section 6 partially overlaps with some parts of our section 3.3. PETR HORAVAˇ 1377 2 K-Theory and Type IIA D-Branes In this section we first review some highlights of [2] (mostly in the context of Type IIB theory), which will give us the opportunity to present some background on K-theory [22]-[26] that will be useful later in the paper.
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